Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 726 · 40 questions

How to solve it (Patterns repeat): Divisibility: 3 and 9 — digit sum; 4 — last two digits; 8 — last three digits; 6 — even and divisible by 3; 11 — alternate digit sums differ by 0 or a multiple of 11. Remainder: find the biggest multiple not more than the number and take it away; to make a number divisible, add divisor − remainder. Units digit of a power: write the cycle (7, 9, 3, 1 for 7), divide the power by the cycle length and use the remainder.

1. 1.Which of these numbers is divisible by 9? (a) 5469 (b) 5485 (c) 5472 (d) 5476
2. 2.Which of these numbers is divisible by 6? (a) 2166 (b) 2174 (c) 2169 (d) 2164
3. 3.Which of these numbers is divisible by 9? (a) 3776 (b) 3781 (c) 3780 (d) 3791
4. 4.Which of these numbers is divisible by 6? (a) 1555 (b) 1547 (c) 1548 (d) 1549
5. 5.What is the units digit of 9^45? (a) 8 (b) 1 (c) 7 (d) 9
6. 6.Which of these numbers is divisible by 4? (a) 2893 (b) 2891 (c) 2888 (d) 2887
7. 7.What is the smallest number you must add to 427 to make it divisible by 9? (a) 4 (b) 6 (c) 7 (d) 5
8. 8.Which of these numbers is divisible by 3? (a) 1327 (b) 1325 (c) 1323 (d) 1322
9. 9.Which of these numbers is divisible by 9? (a) 6638 (b) 6643 (c) 6652 (d) 6642
10. 10.What is the smallest number you must add to 34 to make it divisible by 3? (a) 1 (b) 2
11. 11.What is the units digit of 7^34? (a) 9 (b) 3 (c) 7 (d) 1
12. 12.What is the remainder when 477 is divided by 6? (a) 5 (b) 2 (c) 3 (d) 4
13. 13.Which of these numbers is divisible by 3? (a) 372 (b) 371 (c) 374 (d) 377
14. 14.What is the units digit of 2^10? (a) 8 (b) 6 (c) 4 (d) 2
15. 15.What is the smallest number you must add to 467 to make it divisible by 6? (a) 1 (b) 2 (c) 5 (d) 3
16. 16.What is the smallest number you must add to 114 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
17. 17.What is the units digit of 2^40? (a) 4 (b) 2 (c) 6 (d) 8
18. 18.What is the smallest number you must add to 362 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
19. 19.Which of these numbers is divisible by 4? (a) 300 (b) 298 (c) 301 (d) 307
20. 20.Which of these numbers is divisible by 8? (a) 1061 (b) 1048 (c) 1052 (d) 1047
21. 21.What is the units digit of 3^48? (a) 1 (b) 3 (c) 7 (d) 9
22. 22.Which of these numbers is divisible by 4? (a) 2013 (b) 2012 (c) 2017 (d) 2011
23. 23.Which of these numbers is divisible by 11? (a) 4582 (b) 4557 (c) 4565 (d) 4575
24. 24.Which of these numbers is divisible by 3? (a) 2645 (b) 2647 (c) 2643 (d) 2642
25. 25.Which of these numbers is divisible by 3? (a) 1886 (b) 1883 (c) 1884 (d) 1889
26. 26.What is the smallest number you must add to 493 to make it divisible by 7? (a) 4 (b) 5 (c) 6 (d) 3
27. 27.What is the units digit of 4^30? (a) 7 (b) 5 (c) 4 (d) 6
28. 28.What is the remainder when 791 is divided by 10? (a) 2 (b) 1 (c) 0 (d) 9
29. 29.Which of these numbers is divisible by 4? (a) 2611 (b) 2605 (c) 2615 (d) 2608
30. 30.What is the smallest number you must add to 589 to make it divisible by 10? (a) 1 (b) 9 (c) 2 (d) 3
31. 31.Which of these numbers is divisible by 3? (a) 487 (b) 483 (c) 481 (d) 485
32. 32.What is the smallest number you must add to 261 to make it divisible by 10? (a) 7 (b) 1 (c) 9 (d) 8
33. 33.What is the remainder when 49 is divided by 3? (a) 1 (b) 0 (c) 2
34. 34.Which of these numbers is divisible by 9? (a) 1730 (b) 1720 (c) 1728 (d) 1738
35. 35.What is the smallest number you must add to 183 to make it divisible by 8? (a) 7 (b) 2 (c) 1 (d) 3
36. 36.Which of these numbers is divisible by 11? (a) 8563 (b) 8553 (c) 8558 (d) 8574
37. 37.What is the units digit of 2^58? (a) 6 (b) 2 (c) 8 (d) 4
38. 38.What is the smallest number you must add to 279 to make it divisible by 10? (a) 2 (b) 9 (c) 1 (d) 3
39. 39.What is the smallest number you must add to 473 to make it divisible by 9? (a) 6 (b) 4 (c) 5 (d) 3
40. 40.Which of these numbers is divisible by 8? (a) 2336 (b) 2350 (c) 2333 (d) 2337

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